Requirements for gravity measurements on the anticipated Artemis III mission Peter James1, Anton Ermakov2, and Michael Sori3 1Baylor University 2University of California, Berkeley 3Purdue University August 2020 Contents 1 Introduction 2 2 Measurement requirements for various geological and geophys- ical processes 2 2.1 Science objective #1 { Detect buried volcanic and tectonic struc- tures . 2 2.2 Science objective #2 { Measure porosity in the near subsurface . 4 2.3 Science objective #3 { Detect water ice in the near subsurface . 6 2.4 Science objective #4 { Constrain the regional density of the crust 7 2.5 Science objective #5 { Resolve deep internal structure . 9 2.5.1 Tide-raising potential . 9 2.5.2 Solid body tidal response . 10 2.5.3 Precision with which gravimeter measurements must be made for the tidal response . 12 2.6 Science objective #6 { Detect the solid inner core . 12 arXiv:2009.03514v1 [astro-ph.IM] 8 Sep 2020 1 1 Introduction The purpose of this document is to demonstrate the reasoning behind the specific measurement requirements in the white paper by James et al. titled \The value of surface-based gravity and gravity gradient measurements at the Moon's south pole with Artemis III ". NOTE: As described below, measurement requirements in practice will de- pend on a number of factors, including the geographic location, the shape of the local terrain, the precision to which elevation is known, and the nature of drift in the gravimeter. 2 Measurement requirements for various geo- logical and geophysical processes 2.1 Science objective #1 { Detect buried volcanic and tec- tonic structures A tectonic or volcanic process that produces a change in the distribution of mass underground will generally produce a gravity anomaly. An excess of mass produces a positive \Bouguer gravity anomaly" (defined to be the change in gravity acceleration after taking into account elevation and the mass of topog- raphy), whereas a deficit of mass produces a negative Bouguer gravity anomaly. The amplitude of the gravity anomaly is dependent on the size, geometry, and depth of the mass anomaly. The gravity gradients (i.e., the second derivatives of the gravitational potential) give the spatial rate of change of the vector compo- nents of gravity acceleration. Consequently, gravity gradients are particularly sensitive to shallow, sharp density contrasts. Three such processes are illustrated in figures 1, 2, and 3. Figure 1 illustrates a thrust fault scarp buried under the low-density regolith. For a scarp buried at 100 meters depth with 60 meters of relief, a peak gravity anomaly of roughly 1 mGal is expected. Meanwhile, the Gzz and Gxz components of the gravity gradient tensor produces maximum anomalies of roughly 80 E¨otv¨osrespectively, where an E¨otv¨osis defined to be 10−9s−2. Figure 2 illustrates a series of intrusive volcanic structures. Two 30-meter- wide dikes (i.e., vertical planes filled with solidified magma) are included, along with one 200-meter-thick sill (i.e., a horizontal plane filled with solidified magma) at one kilometer depth. For intrusive magmatic bodies with these geometries and a density contrast of 700 kg / m3 with respect to the surrounding crust, the maximum gravity anomaly is a little more than 3 mGals. The corresponding gravity gradients are maximized at roughly 40 E¨otv¨osrespectively. Finally, figure 3 illustrates a lava tube with a width of one kilometer and a height of 250 meters. Since lava tubes are void spaces, they represent a signifi- cant deficit of mass and produce a large negative Bouguer gravity anomaly. The maximum-amplitude gravity anomaly is roughly -18 mGals. The corresponding gravity gradients are maximized at roughly 300 E¨otv¨osrespectively. 2 100 Gzz 50 Gxz 0 -50 -100 gradient (Eötvös) gradient -1000 -800 -600 -400 -200 0 200 400 600 800 1000 1 0.5 gravity (mGal) gravity 0 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 distance (meters) ρ = 1700 kg/m3 60 m ρ = 2600 kg/m3 Figure 1: The gravity gradients and gravity anomaly that would be generated by a tectonic scarp buried under the lunar regolith. 50 Gzz Gxz 0 -50 gradient (Eötvös) gradient -5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000 4 3 2 1 gravity (mGal) gravity 0 -5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000 distance (meters) ρ = 2500 kg/m3 30 m 1 km dike ρ = 3200 kg/m3 Figure 2: The gravity gradients and gravity anomaly that would be generated by a pair of 30-meter-wide intrusive volcanic dikes connected by a 200-meter-tall sill. 3 400 Gzz 200 Gxz 0 -200 -400 gradient (Eötvös) gradient -1000 -800 -600 -400 -200 0 200 400 600 800 1000 0 -5 -10 -15 gravity (mGal) gravity -20 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 distance (meters) ρ = 2200 kg/m3 250 m ρ = 0 kg/m3 Figure 3: The gravity gradients and gravity anomaly that would be generated by a lava tube that is roughly 10% of the maximum theoretical width or 1% of the maximum theoretical cross-sectional area (Blair et al., 2017). While lava tubes are suspected to exist in various locations in the lunar maria, lava tubes are not known to exist in the vicinity of the expected landing sites for Artemis III. Consequently, we will take our measurement requirements from the other two test problems illustrated here. The minimum gravimeter resolution that would plausibly identify near-surface features of geologic interest would be 3 mGals (cf. Figure 2). The minimum gravity gradiometer resolution that would plausibly identify near-surface features of geologic interest would be 100 E¨otv¨os (cf. Figure 1). 2.2 Science objective #2 { Measure porosity in the near subsurface When gravity is measured on short baselines with undulating topography, the resulting data may be used to estimate the bulk density of the lunar regolith. While the bulk density of the lunar regolith is somewhat dependent on the composition of the regolith, the bulk density is primarily determined by the porosity of the regolith: ρbulk = (1 − φ)ρgrain (1) 4 where φ is porosity, ρbulk is the bulk density, and ρgrain is the average grain density of the regolith particles. In practice, the density of a terrain may be measured using the method of Nettleton (1939). In this technique, gravity is measured at a variety of elevations in a rolling terrain. Short-wavelength topography is generally independent of geology in the deeper crust, in which case the Bouguer anomaly would not be correlated with topography. Therefore, when the proper bulk density for the near subsurface is chosen, the resulting Bouguer gravity anomaly should become uncorrelated with topography. This process is illustrated in Figure 4. Bulk density varies significantly with depth in the lunar interior (Besserer et al., 2014), and the effective depth-sensitivity of a Nettleton-type analysis is dependent on the wavelength of the topography. When gravity from harmonic topography is upward continued by a distance z, that signal is attenuated by a factor of exp(−2πz/λ) where λ is the wavelength. Therefore the characteristic sensitivity depth zs is equal to λ/2π. For example, gravity collected over topog- raphy with a wavelength of 100 meters would be sensitive to the uppermost 16 meters of the subsurface. Measurement objectives for this scientific investigation are governed by the precision with which we would like to measure the bulk density of the near sub- surface. We will arbitrarily aim to estimate bulk density with the same precision as Lewis et al. (2019) { namely, 180 kg/m3. The largest source of variation in gravitational acceleration corresponds to the distance from the Moon's center of mass, so the first step in interpreting a set of raw gravity readings is to calculate the \free-air gravity anomaly", which is the residual gravity acceleration after accounting for the elevation of the measurement. The free-air gravity difference ∆g corresponding to an elevation difference ∆h is, to first order, predicted by a Bouguer slab (i.e., an infinite horizontal slab) between these two elevations. In practice, this allows us to calculate bulk density: ∆g ρ = (2) 2πG∆h This may be further broken down by noting that the raw gravity reading is converted to Bouguer gravity through a topographic correction. The Moon's free air gradient is 0.18 mGal/m, which must be added to a raw gravity reading: ∆g + 0:18∆h ρ = raw (3) 2πG∆h or ∆g 0:18 ρ = raw + (4) 2πG∆h 2πG The error in the density measurement is then determined by the error in the raw gravity reading, E(∆graw), as well as the error in the elevations, E(∆h): E(∆g ) ∆g E(∆h) E(ρ) ≈ raw + raw · (5) 2πG∆h 2πG∆h ∆h 5 3 1,600 kg/m 1,800 y anomaly t vi 2,200 a r Figure 4: An illustration of Nettleton's g 2,400 method for determining the bulk den- 2,600 sity of a terrain (modified from Hinze ouguer et al. (2013)). For this example, the B density of the terrain has been deter- mined to be 2200 kg/m3 since that is the tion va density that de-correlates the Bouguer Ele gravity anomaly (top) from the topog- Distance raphy (bottom). From the first term, we can see that if a survey traverses 20 meters of eleva- tion, 0.1 mGal of error in the gravity readings would translate to 159 kg/m3 of error in the calculated density. The second term shows that the error in the density estimate is roughly proportional to the fractional error in elevation.